Question

The sum of co-efficients of all even degree terms in $x$ in the expansion of $(x+\sqrt{x^3-1}{)}^6+(x-\sqrt{x^3-1}{)}^6,(x>1)$ is equal to:

• A. $24$
• B. $26$
• C. $29$
• D. $32$

Answer 1

The correct option is A $24$

If we add terms of both the expansions, even numbered terms will cancel each other and we get-
$2[{}^6{C}_0{x}^6+{}^6{C}_2{x}^4(\sqrt{x^3-1}{)}^2+{}^6{C}_4{x}^2(\sqrt{x^3-1}{)}^4+{}^6{C}_6(\sqrt{x^3-1}{)}^6]=2[{}^6{C}_0{x}^6+{}^6{C}_2{x}^4(x^3-1)+{}^6{C}_4{x}^2(x^3-1{)}^2+{}^6{C}_6(x^3-1{)}^3]=2[x^6+{}^6{C}_2{x}^7-{}^6{C}_2{x}^4+{}^6{C}_4{x}^8+{}^6{C}_4{x}^2-2{}^6{C}_4{x}^5+x^9-1-3x^6+3x^3]$

Hence, sum of coefficients of even powers of $x$
$=2[1-{}^6{C}_2+{}^6{C}_4+{}^6{C}_4-1-3]=2[1-15+15+15-4]=24$